Quite how big will depend on the outcome of a lawsuit filed by the US Securities and Exchange Commission on Thursday.
It accuses Mr Musk of securities fraud after tweeting that he had “funding secured” to take Tesla private, sending the stock price soaring.
Mr Musk was lying about having that funding, the SEC said - although Mr Musk has called the lawsuit “unjustified”.
The worst case scenario for Mr Musk is that he pays an enormous fine, and is banned from being a director at any public company in the US. That would mean his days as Tesla’s chief executive would be over - an almost unthinkable prospect for a company intrinsically linked to Mr Musk’s personality and vision for the future of clean mobility.
"I think the important issue [for Tesla] will be to fashion a remedy that simultaneously disciplines Mr Musk, but without destroying his value to Tesla’s shareholders,” said Professor Joseph Grundfest from Stanford Law School, and a former SEC commissioner.
Get a ‘Twitter nanny'
"There’s a spectrum of potential solutions,” Prof Grundfest told the BBC.
"You could imagine, in one extreme, he becomes chief product officer and someone takes over as CEO.
"Or, he gets what I call a 'Twitter nanny', where he can’t communicate without first clearing it with a responsible adult.”
A Tesla spokeswoman did not wish to comment on Mr Musk's future.
Prof Grundfest likened the situation to that of American businesswoman Martha Stewart, who in 2004 was found guilty of insider trading.
As well as serving jail time, she agreed to a five-year ban from serving as a director at her company. Instead, she took up a role as chief creative officer for that period.
In Mr Musk’s case, the SEC is not pushing for jail time.
Reflecting on the saga, Prof Grundfest added: "Let’s face it. America leads the world in stupid tweets.”
Another question is how the case affects Mr Musk’s reputation among his customers.
Reacting to the news of the lawsuit, Tesla’s famously devoted fans have been left frustrated, angry and worried about the company’s future.
As well as the SEC lawsuit, the Department of Justice is in the “early stages” of its own investigation. On top of that, Mr Musk is facing a defamation case after he continually, on Twitter and in emails, accused a British diving expert of being a paedophile.
The Tesla community, which has typically leapt to the 47-year-old’s defence in the face of negative press, is showing growing concern about the distractions created by Mr Musk’s actions.
"If only Elon Musk would not keep shooting off at the mouth like a stupid narcissist,” wrote user Avip, on the Tesla Motors Club forum.
Another user, rolosrevenge, added: "This was a stupid, self inflicted wound by Elon."
More about: ElonMusk